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We describe how traceless projection of tensors of a given rank can beconstructed in a closed form. On the way to this goal we invoke therepresentation theory of the Brauer algebra and the related Schur-Weyldualities. The resulting traceless projector is constructed from purelycombinatorial data involving Young diagrams. By construction, the projectormanifestly commutes with the symmetric group and is well-adapted torestrictions to $GL$-irreducible tensor representations. We develop auxiliarycomputational techniques which serve to take advantage of the obtained resultsfor applications. The proposed method of constructing traceless projectorsleads to a particular central idempotent in the semisimple regime of the Braueralgebra.

Construction of the traceless projection of tensors via the Brauer algebra